I have an interview with a Harvard alum tomorrow at a Starbucks. What should I order and who should pay? I know this is the least of my worries, but this is how my brain works haha.
order what you like (coffee, tea, juice etc.) and have the expectation that you’ll be paying for yourself.
Come earlier and get a cup of coffee or whatever. Don’t get food unless you expect to finish it before the interviewer arrives. They’ll get what they want.
I have interviewed Harvard applicants for about 5 years, all at Starbucks. 
One thing you don’t want to do is to offer to purchase anything for the interviewer. I’ve never had a student who offered to purchase a cup of whatever, and I’m glad that no one has. If the interviewer volunteers to purchase something for you, on the other hand, you can politely decline or accept the offer if the situation calls for it, i.e., the interviewer really wants to get you something.
Either way, getting something to drink allows both the interviewer and especially the student relax more. It’s not a formal job interview. Consider it as an informal session to get to know one another, so have fun in the process.
I do alumni interviews for a different university, and always meet at Starbucks. I generally get there first and order a drink for myself. When the student arrives, I always ask if they would like to get something while I wait. Most have opted not to get something or have said they will get one on the way out.
Drinking is fine, but I think eating on both parts would be awkward.
Good luck to you.